MATH 201 - INTRODUCTION TO PROBABILITY AND STATISTICS Combinatorial Analysis Question Bank Set 8 | MATH 201 - Introduction to Probability and Statistics

MATH 201 - INTRODUCTION TO

PROBABILITY AND STATISTICS -

Combinatorial Analysis

Question Bank - Set 8

Liberty University

Question 1

Question

In a group of 10 students, how many ways can we form a committee of 3 stu-

dents?

Solution

Step 1: To solve this problem, we will use the formula for combinations, which

is given by:

C(n, k) = n!

k!(n−k)!

Step 2: In this case, we have 10 students and we want to choose a committee

of 3 students. So, we have:

C(10,3) = 10!

3!(10 −3)!

Step 3: Calculating the factorials in the formula, we get:

C(10,3) = 10 ×9×8

3×2×1

Step 4: Simplifying the expression, we find:

C(10,3) = 120

Step 5: Therefore, there are 120 different ways to form a committee of 3

students from a group of 10 students.

Question 2

Question

A committee of 5 people is to be formed from a group of 9 students and 7

teachers. In how many ways can the committee be formed if it must have at

least 3 students?

Solution

Step 1: Calculate the number of ways to select a committee with exactly 3

students, 2 teachers. The number of ways to select 3 students out of 9 is

3)=9!

3!(9−3)! = 84. The number of ways to select 2 teachers out of 7 is

2)=7!

2!(7−2)! = 21. So, the total number of ways to select a committee with

exactly 3 students, 2 teachers is 84 ×21 = 1764.

Step 2: Calculate the number of ways to select a committee with exactly

4 students, 1 teacher. The number of ways to select 4 students out of 9 is

4)=9!

4!(9−4)! = 126. The number of ways to select 1 teacher out of 7 is

1)=7!

1!(7−1)! = 7. So, the total number of ways to select a committee with

exactly 4 students, 1 teacher is 126 ×7 = 882.

Step 3: Calculate the number of ways to select a committee with exactly

5 students, 0 teachers. The number of ways to select 5 students out of 9 is

5)=9!

5!(9−5)! = 126. Since no teachers are allowed in this committee, the

number of ways to select 0 teachers out of 7 is (7

0)= 1. So, the total number of

ways to select a committee with exactly 5 students, 0 teachers is 126 ×1 = 126.

Step 4: Add up the totals from Step 1, Step 2, and Step 3 to get the final

answer. Total number of ways to form the committee with at least 3 students

=1764 + 882 + 126 = 2772. Therefore, the committee can be formed in 2772

ways.

Question 3

Question

In a group of 8 people, how many ways can you form a committee of 3 people?

Solution

To find the number of ways to form a committee of 3 people from a group of 8,

we can use the formula for combinations:

Number of ways to choose kitems from nitems =(n

k)=n!

k!(n−k)!

Step 1: Calculate the number of ways to choose 3 people from 8:

3)=8!

3!(8 −3)! =8×7×6

3×2×1= 56

Step 2: Therefore, there are 56 ways to form a committee of 3 people from

a group of 8.

Question 4

Question

A committee of 5 people is to be formed from a group of 8 students and 3

professors. How many ways can this committee be formed if it must consist of

3 students and 2 professors?

Solution

Step 1: Calculate the number of ways to choose 3 students from 8. There are

3)ways to choose 3 students from a group of 8.

3)=8!

3!(8 −3)! =8×7×6

3×2×1= 56

Step 2: Calculate the number of ways to choose 2 professors from 3. There

are (3

2)ways to choose 2 professors from a group of 3.

2)=3!

2!(3 −2)! =3

2= 3

Step 3: Multiply the results from Step 1 and Step 2 to find the total number

of ways the committee can be formed.

56 ×3 = 168

Therefore, there are 168 ways to form a committee of 5 people consisting of

3 students and 2 professors from a group of 8 students and 3 professors.

Question 5

Question

A committee of 5 people is to be formed from a group of 10 men and 8 women.

In how many ways can the committee be formed if it must consist of at least 2

men and 2 women?

Solution

Step 1: Find the number of ways to choose 2 men and 3 women from the group.

There are (10

2)ways to choose 2 men from 10 men, and (8

3)ways to choose

3 women from 8 women.

Therefore, the number of ways to choose 2 men and 3 women is (10

2)×(8

3).

Step 2: Find the number of ways to choose 3 men and 2 women from the

group.

There are (10

3)ways to choose 3 men from 10 men, and (8

2)ways to choose

2 women from 8 women.

Therefore, the number of ways to choose 3 men and 2 women is (10

3)×(8

2).

Step 3: Add the results of Step 1 and Step 2 to find the total number of

ways to form the committee with at least 2 men and 2 women.

The total number of ways is (10

2)×(8

3)+(10

3)×(8

2).

Calculating these values, we get:

(10

2)= 45,(8

3)= 56,(10

3)= 120,(8

2)= 28

So the total number of ways to form the committee is 45 ×56 + 120 ×28 =

2520 + 3360 = 5880. Thus, the committee can be formed in 5880 ways.

Question 6

Question

A committee of 5 members is to be formed from a group of 8 men and 6 women.

In how many ways can the committee be formed if it must consist of at least 2

women?

Solution

Step 1: Calculate the number of ways to choose 2 women out of 6. There are

2)ways to choose 2 women from a group of 6. (6

2)=6!

2!(6−2)! = 15 ways.

Step 2: Calculate the number of ways to choose the remaining members from

the men and women. Since the committee must consist of at least 2 women, we

can choose the remaining members (men and/or women) from the remaining

total of 8 men and 4 women. We can choose the remaining 3 members from a

group of 8 men and 4 women in (12

3)ways. (12

3)=12!

3!(12−3)! = 220 ways.

Step 3: Find the total number of ways to form the committee. Multiply

the number of ways to choose the 2 women from Step 1 by the number of

ways to choose the remaining members from Step 2. Total number of ways =

15 ×220 = 3300.

Therefore, the committee can be formed in 3300 ways if it must consist of

at least 2 women.

Question 7

Question

In a particular city, there are 4 different train stations. A commuter needs

to travel from one station to another station for work. If the commuter al-

ways chooses the shortest route, how many different routes are possible for the

commuter to take?

Solution

Step 1: Since there are 4 different train stations, the commuter can choose the

starting station in 4 ways.

Step 2: After choosing the starting station, the commuter needs to choose

the destination station. Since the commuter always chooses the shortest route,

there is only 1 shortest route from any starting station to any destination station.

Step 3: Therefore, the total number of possible routes the commuter can

take is the product of the number of ways to choose the starting station and the

number of ways to choose the destination station.

Step 4: This gives us 4×1 = 4 possible routes that the commuter can take.

Hence, there are 4 different routes possible for the commuter to take when

traveling from one station to another station in the city.

Question 8

Question

In a group of 10 students, how many ways are there to select a committee of 5

students if exactly 2 of the students must be included?

Solution

Step 1: Calculate the number of ways to choose 2 out of the 10 students.

There are (10

2)ways to choose 2 students from a group of 10.

Step 2: Calculate the number of ways to choose the remaining 3 students

from the 8 students left after selecting the required 2 students.

There are (8

3)ways to choose 3 students from a group of 8.

Step 3: Multiply the results from Step 1 and Step 2 to find the total number

of ways to form the committee.

Multiplying (10

2)and (8

3)gives the total number of ways to choose a com-

mittee of 5 students.

Therefore, there are (10

2)×(8

3)ways to select a committee of 5 students where

exactly 2 students must be included.

Question 9

Question

In how many ways can 5 girls and 3 boys be arranged in a row if the girls must

sit together and the boys must sit together?

Solution

Step 1: Treat the girls as one group and the boys as another group. Since the

girls must sit together and the boys must sit together, we treat the two groups

as one combined group. Step 2: We now have 1 group of 8 people to arrange in

a row. There are 8! ways to arrange the 8 people. Step 3: Within each group

(girls and boys), they can be arranged among themselves in a certain number

of ways. For the 5 girls, there are 5! ways to arrange them, and for the 3 boys,

there are 3! ways to arrange them. Step 4: Therefore, the total number of ways

the 5 girls and 3 boys can be arranged in a row if the girls must sit together

and the boys must sit together is 8! ×5! ×3!.

Question 10

Question

A committee of 3 students is to be chosen from a group of 10 students. If

there are 4 freshmen, 3 sophomores, 2 juniors, and 1 senior, how many different

possible committees can be formed if the committee must contain at least one

student from each class?

Solution

Step 1: Calculate the number of ways to choose one student from each class.

Step 2: Calculate the number of ways to choose the remaining student.

Step 1: There are 4 freshmen, 3 sophomores, 2 juniors, and 1 senior. We

need to choose one student from each class, so the number of ways to do this is:

4×3×2×1 = 24

Step 2: After choosing one student from each class, we need to select one

more student from the remaining 6 students to complete the committee. Since

there are no restrictions on choosing the last student, the number of ways to do

this is: (6

1)= 6

Therefore, the total number of different possible committees that can be

formed is:

24 ×6 = 144

Question 11

Question

A committee of 5 students needs to be formed out of a group of 10 math majors

and 8 engineering majors. In how many ways can this committee be formed if

it must consist of 3 math majors and 2 engineering majors?

Solution

Step 1: Calculate the number of ways to choose 3 math majors from 10 students.

There are (10

3)=10!

3!(10−3)! = 120 ways to choose 3 math majors.

Step 2: Calculate the number of ways to choose 2 engineering majors from

8 students. There are (8

2)=8!

2!(8−2)! = 28 ways to choose 2 engineering majors.

Step 3: Find the total number of ways to form the committee. Since the

committee must consist of 3 math majors and 2 engineering majors, we multiply

the number of ways to choose math majors by the number of ways to choose

engineering majors: Total number of ways = 120 ×28 = 3360.

Therefore, there are 3360 ways to form a committee of 5 students with 3

math majors and 2 engineering majors.

Question 12

Question

A committee of 5 people is to be formed from a group of 9 individuals (4 men

and 5 women). How many different committees can be formed if the committee

must consist of at least 2 women?

Solution

Step 1: Calculate the number of ways to choose a committee with exactly 2

women, 3 women, 4 women, or 5 women.

For exactly 2 women and 3 men: There are 5 ways to choose 2 women out

of 5 women, and 4 ways to choose 3 men out of 4 men. Therefore, the number

of ways to choose a committee with exactly 2 women and 3 men is:

5×4 = 20

For exactly 3 women and 2 men: There are 5 ways to choose 3 women out

of 5 women, and 4 ways to choose 2 men out of 4 men. Therefore, the number

of ways to choose a committee with exactly 3 women and 2 men is:

5×6 = 30

For exactly 4 women and 1 man: There are 5 ways to choose 4 women out

of 5 women, and 4 ways to choose 1 man out of 4 men. Therefore, the number

of ways to choose a committee with exactly 4 women and 1 man is:

5×4 = 20

For 5 women and 0 men: There is only 1 way to choose all 5 women. There-

fore, the number of ways to choose a committee with 5 women and 0 men is:

Step 2: Add up the number of ways from each case to find the total number

of different committees that can be formed.

20 + 30 + 20 + 1 = 71

Therefore, there are 71 different committees that can be formed from the

group of 9 individuals where the committee must consist of at least 2 women.

Question 13

Question

A committee of 5 people is to be formed from a group of 7 men and 4 women.

In how many ways can the committee be formed if it must consist of at least 2

women?

Solution

Step 1: Calculate the number of ways to form a committee with 2 women and

3 men. Since there are 4 women and 7 men, the number of ways to choose 2

women out of 4 and 3 men out of 7 is given by the binomial coeﬀicient:

2)×(7

3)= 6 ×35 = 210.

Step 2: Calculate the number of ways to form a committee with 3 women

and 2 men. Similarly, the number of ways to choose 3 women out of 4 and 2

men out of 7 is given by the binomial coeﬀicient:

3)×(7

2)= 4 ×21 = 84.

Step 3: Calculate the total number of ways to form the committee with at

least 2 women. The total number of ways to form the committee with at least

2 women is the sum of the ways in Step 1 and Step 2:

210 + 84 = 294.

Therefore, there are 294 ways to form the committee with at least 2 women.

Question 14

Question

In a certain country, license plates for cars consist of 3 letters followed by 3

digits. The first letter must be a vowel (A, E, I, O, U) and the letters cannot

be repeated. How many different license plates can be created following these

rules?

Solution

Step 1: Determine the number of ways to choose the first letter from the set of

vowels. There are 5 vowels (A, E, I, O, U) to choose from for the first letter.

Therefore, there are 5 choices for the first letter.

Step 2: Determine the number of ways to choose the second letter from the

remaining set of 20 consonants. After choosing the first letter, there are 20

letters left for the second position (26 total letters - 1 used vowel). Therefore,

there are 20 choices for the second letter.

Step 3: Determine the number of ways to choose the third letter from the

remaining set of 19 consonants. After choosing the first two letters, there are 19

letters left for the third position. Therefore, there are 19 choices for the third

letter.

Step 4: Determine the number of ways to choose the first digit. There are

10 digits (0-9) to choose from for the first digit. Therefore, there are 10 choices

for the first digit.

Step 5: Determine the number of ways to choose the second digit. After

choosing the first digit, there are 9 digits left for the second digit. Therefore,

there are 9 choices for the second digit.

Step 6: Determine the number of ways to choose the third digit. After

choosing the first two digits, there are 8 digits left for the third digit. Therefore,

there are 8 choices for the third digit.

Step 7: Multiply the number of choices together to find the total number of

different license plates that can be created. Total number of license plates = 5

(choices for the first letter) ×20 (choices for the second letter) ×19 (choices

for the third letter) ×10 (choices for the first digit) ×9 (choices for the second

digit) ×8 (choices for the third digit) Total number of license plates = 5×20 ×

19 ×10 ×9×8 = 1,368,000

Therefore, there are 1,368,000 different license plates that can be created

following the given rules.

Question 15

Question

How many different ways can you arrange the letters in the word ”STATISTICS”

if the two S’s are next to each other?

Solution

Step 1: Treat the two S’s as one entity. Step 2: The word ”STATISTICS” now

has 9 entities - S, T, A, I, S, T, I, C, S (treating the two S’s as one entity).

Step 3: There are 9! ways to arrange these entities. Step 4: However, the two

S’s can be arranged in 2! ways. Step 5: Therefore, the total number of ways to

arrange the letters in the word ”STATISTICS” if the two S’s are next to each

other is 9!

2! or 181,440 ways.

Question 16

Question

In how many ways can 5 different couples be seated at a round table if each

couple must sit together?

Solution

To solve this problem, we can treat each couple as a single entity. Then we have

5 entities to arrange around a circle (since it’s a round table).

Step 1: First, we arrange the 5 entities (couples) around the circle.

There are (5 −1)! = 4! = 24 ways to arrange 5 different entities around a

circle.

Step 2: Next, within each entity (couple), there are 2 ways to arrange the

individuals.

Since there are 5 entities, each of which has 2 ways to be arranged, the total

number of ways to arrange the individuals within the couples is 25= 32.

Step 3: Finally, we multiply the results from Step 1 and Step 2 to find the

total number of ways.

Total number of ways = 24 ×32 = 768.

Therefore, there are 768 ways to seat 5 different couples at a round table

if each couple must sit together.

Question 17

Question

In a group of 8 friends, how many ways can 3 of them be selected to form a

committee?

Solution

Step 1: To solve this problem, we will use the concept of combinations. The

number of ways to choose kobjects from a set of nobjects is given by the

formula:

C(n, k) = n!

k!(n−k)!

where n!denotes the factorial of n, which is the product of all positive

integers up to n.

Step 2: In this problem, we have 8 friends and we need to select 3 of them

to form a committee. So, we have n= 8 and k= 3.

Step 3: Substituting n= 8 and k= 3 in the formula for combinations, we

get:

C(8,3) = 8!

3!(8 −3)!

Step 4: Calculating the factorials, we get:

C(8,3) = 8×7×6

3×2×1= 56

Step 5: Therefore, there are 56 ways to select 3 friends from a group of 8

friends to form a committee.

Question 18

Question

A committee of 5 people is to be formed from a group of 8 men and 6 women.

If at least 2 women must be on the committee, how many different committees

can be formed?

Solution

Step 1: First, let’s consider the scenario where exactly 2 women are on the

committee. - Choose 2 women out of 6: (6

2)= 15 ways - Choose 3 men out of 8:

3)= 56 ways - Multiply the number of ways for women and men: 15×56 = 840

ways

Step 2: Next, let’s consider the scenario where exactly 3 women are on the

committee. - Choose 3 women out of 6: (6

3)= 20 ways - Choose 2 men out of 8:

2)= 28 ways - Multiply the number of ways for women and men: 20×28 = 560

ways

Step 3: Finally, let’s consider the scenario where all 6 women are on the

committee. - Choose all 6 women out of 6: (6

6)= 1 way - Choose 3 men out of

8: (8

3)= 56 ways - Multiply the number of ways for women and men: 1×56 = 56

ways

Step 4: Add up the number of committees from each scenario to get the

total number of committees: 840 + 560 + 56 = 1456 ways

Therefore, there are 1456 different committees that can be formed.

Question 19

Question

In a committee of 8 members, there are 4 professors and 4 students. If the

committee needs to select a president and a vice president, how many ways can

this be done if the president must be a professor and the vice president must be

a student?

Solution

Step 1: Selecting the president from the 4 professors can be done in 4 ways.

Step 2: Selecting the vice president from the 4 students can be done in 4

ways.

Step 3: To find the total number of ways to select both a president and a

vice president, we multiply the number of ways from Step 1 and Step 2.

Therefore, the total number of ways to select a president and a vice president

as specified is 4×4 = 16 .

Question 20

Question

A committee of 5 people is to be formed from a group of 8 men and 6 women.

In how many ways can this committee be formed if it must contain at least 2

women?

Solution

Step 1: Calculate the total number of ways to form a committee of 5 people

from the group of 14 individuals.

There are 14 individuals in total (8 men and 6 women) from which we need

to choose 5 people to form the committee. This can be calculated using the

combination formula (n

k)=n!

k!(n−k)! , where nis the total number of individuals

and kis the number of individuals to be chosen.

Therefore, the total number of ways to form the committee is:

(14

5)=14!

5!(14 −5)! =14!

5!9! = 2002

Step 2: Calculate the number of ways to form a committee with at most 1

woman.

For the committee to contain at least 2 women, we need to find the number

of ways to form a committee with at most 1 woman and then subtract this from

the total number of ways to get the desired result.

Number of ways to form a committee with 0 women:

5)=8!

5!(8 −5)! = 56

Number of ways to form a committee with 1 woman:

1)×(8

4)=(6

1)×(8

8−4)= 6 ×70 = 420

Therefore, the total number of ways to form a committee with at most 1

woman is:

56 + 420 = 476

Step 3: Calculate the number of ways to form a committee with at least 2

women.

The number of ways to form a committee with at least 2 women is given by:

2002 −476 = 1526

Hence, there are 1526 ways to form a committee of 5 people that must

contain at least 2 women from the group of 8 men and 6 women.

Question 21

Question

A committee of 5 members is to be formed from a group of 10 men and 8 women.

How many ways can the committee be formed if it must have at least 3 women?

Solution

Step 1: Calculate the number of ways to form a committee with exactly 3

women.

• Choose 3 women out of 8: (8

• Choose 2 men out of 10: (10

• Multiply the two choices together to get the total number of ways to form

a committee with exactly 3 women: (8

3)×(10

Step 2: Calculate the number of ways to form a committee with exactly 4

women.

• Choose 4 women out of 8: (8

• Choose 1 man out of 10: (10

• Multiply the two choices together to get the total number of ways to form

a committee with exactly 4 women: (8

4)×(10

Step 3: Calculate the number of ways to form a committee with exactly 5

women.

• Choose 5 women out of 8: (8

• No men are needed in this case.

• The total number of ways to form a committee with exactly 5 women is

simply (8

Step 4: Add up the total number of ways to form a committee with at least

3 women.

• Total ways = ways with exactly 3 women + ways with exactly 4 women

+ ways with exactly 5 women

• Total ways = (8

3)×(10

2)+(8

4)×(10

1)+(8

Therefore, the total number of ways to form a committee with at least 3

women is (8

3)×(10

2)+(8

4)×(10

1)+(8

5).

Question 22

Question

A committee of 3 students is to be formed from a group of 10 students. If two

of the students cannot serve together, how many different committees can be

formed?

Solution

Step 1: First, let’s find the total number of ways to choose a committee of 3

students from a group of 10 students. This is simply the combination formula,

which is given by:

Total number of ways =(10

3)=10!

3!(10 −3)! =10 ×9×8

3×2×1= 120.

Step 2: Now, let’s find the number of ways to form a committee where the

two specific students cannot serve together. We can find this by subtracting

the number of committees that include the two specific students from the total

number of committees.

Step 3: Let’s calculate the number of ways the two specific students can

serve together. This can be done by forming a committee of 2 students from

the remaining 8 students (excluding the two specific students), and then adding

the two specific students to the committee. This is given by:

1)×(2

2)=(8

1)= 8.

Step 4: Therefore, the number of committees where the two specific students

cannot serve together is:

120 −8 = 112.

So, there are 112 different committees that can be formed from the group of

10 students where the two specific students cannot serve together.

Question 23

Question

In how many ways can you choose a committee of 4 people from a group of 10

individuals?

Solution

Step 1: To solve this problem, we will use the combination formula, which is

given by:

C(n, r) = n!

r!(n−r)!

where nis the total number of individuals, ris the number of people we want

to choose, and n!denotes the factorial of n(the product of all positive integers

up to n).

Step 2: Substituting n= 10 and r= 4 into the formula, we get:

C(10,4) = 10!

4!(10 −4)!

Step 3: Calculating the factorials in the formula:

10! = 10 ×9×8×7×6×5×4×3×2×1 = 3628800

4! = 4 ×3×2×1 = 24

6! = 6 ×5×4×3×2×1 = 720

Step 4: Substituting the factorials back into the formula:

C(10,4) = 3628800

24 ×720 =3628800

17280 = 210

Step 5: Therefore, there are 210 ways to choose a committee of 4 people

from a group of 10 individuals.

Question 24

Question

A committee of 5 people is to be formed from a group of 7 men and 8 women.

Calculate the number of ways the committee can be formed such that there are

at least 3 women on the committee.

Solution

Step 1: Calculate the total number of ways to form a committee of 5 people

from 15.

There are a total of 15 people to choose from, and we need to choose a

committee of 5. This can be calculated using the combination formula (n

r)=

r!(n−r)! , where nis the total number of people and ris the number of people

we are choosing.

So, the total number of ways to form a committee of 5 from 15 is:

(15

5)=15!

5!(15 −5)!

Step 2: Calculate the number of ways to form a committee with at most 2

women.

We can first calculate the number of ways to form a committee with 0 women

(all men) and the number of ways to form a committee with 1 or 2 women.

Number of ways to form a committee with 0 women:

Number of ways to form a committee with 1 woman:

1)×(7

Number of ways to form a committee with 2 women:

2)×(7

So, the total number of ways to form a committee with at most 2 women is:

5)+(8

1)×(7

4)+(8

2)×(7

Step 3: Calculate the number of ways to form a committee with at least 3

women.

The total number of ways to form a committee with at least 3 women is the

complement of the number of ways to form a committee with at most 2 women.

Therefore, the number of ways to form a committee with at least 3 women is:

(15

5)−((7

5)+(8

1)×(7

4)+(8

2)×(7

3))

Question 25

Question

A committee of 4 people is to be formed from a group of 10 students. How

many ways can this committee be formed if there are 3 specific students that

must be included in it?

Solution

Step 1: Choose the 3 specific students to be included in the committee. There

are (10

3)ways to choose the 3 specific students out of 10.

Step 2: Choose the remaining 1 student to be included in the committee.

After selecting the 3 specific students, there are 7 students remaining to choose

from. Thus, there are (7

1)ways to select 1 student from the remaining 7.

Step 3: Multiply the number of ways from steps 1 and 2 to determine the

total number of ways to form the committee. Therefore, the total number of

ways to form the committee with 3 specific students is:

(10

3)×(7

1)= 120 ×7 = 840

Hence, there are 840 ways to form a committee of 4 people from a group of

10 students with 3 specific students included.

Question 2

Question

A committee of 5 people is to be formed from a group of 9 students and 7

teachers. In how many ways can the committee be formed if it must have at

least 3 students?

Solution

Step 1: Calculate the number of ways to select a committee with exactly 3

students, 2 teachers. The number of ways to select 3 students out of 9 is

3)=9!

3!(9−3)! = 84. The number of ways to select 2 teachers out of 7 is

2)=7!

2!(7−2)! = 21. So, the total number of ways to select a committee with

exactly 3 students, 2 teachers is 84 ×21 = 1764.

Step 2: Calculate the number of ways to select a committee with exactly

4 students, 1 teacher. The number of ways to select 4 students out of 9 is

4)=9!

4!(9−4)! = 126. The number of ways to select 1 teacher out of 7 is

1)=7!

1!(7−1)! = 7. So, the total number of ways to select a committee with

exactly 4 students, 1 teacher is 126 ×7 = 882.

Step 3: Calculate the number of ways to select a committee with exactly

5 students, 0 teachers. The number of ways to select 5 students out of 9 is

5)=9!

5!(9−5)! = 126. Since no teachers are allowed in this committee, the

number of ways to select 0 teachers out of 7 is (7

0)= 1. So, the total number of

ways to select a committee with exactly 5 students, 0 teachers is 126 ×1 = 126.

Step 4: Add up the totals from Step 1, Step 2, and Step 3 to get the final

answer. Total number of ways to form the committee with at least 3 students

=1764 + 882 + 126 = 2772. Therefore, the committee can be formed in 2772

ways.

Question 3

Question

In a group of 8 people, how many ways can you form a committee of 3 people?

Solution

To find the number of ways to form a committee of 3 people from a group of 8,

we can use the formula for combinations:

Number of ways to choose kitems from nitems =(n

k)=n!

k!(n−k)!

Step 1: Calculate the number of ways to choose 3 people from 8:

3)=8!

3!(8 −3)! =8×7×6

3×2×1= 56

Step 2: Therefore, there are 56 ways to form a committee of 3 people from

a group of 8.

Question 4

Question

A committee of 5 people is to be formed from a group of 8 students and 3

professors. How many ways can this committee be formed if it must consist of

3 students and 2 professors?

Solution

Step 1: Calculate the number of ways to choose 3 students from 8. There are

3)ways to choose 3 students from a group of 8.

3)=8!

3!(8 −3)! =8×7×6

3×2×1= 56

Step 2: Calculate the number of ways to choose 2 professors from 3. There

are (3

2)ways to choose 2 professors from a group of 3.

2)=3!

2!(3 −2)! =3

2= 3

Step 3: Multiply the results from Step 1 and Step 2 to find the total number

of ways the committee can be formed.

56 ×3 = 168

Therefore, there are 168 ways to form a committee of 5 people consisting of

3 students and 2 professors from a group of 8 students and 3 professors.

Question 5

Question

A committee of 5 people is to be formed from a group of 10 men and 8 women.

In how many ways can the committee be formed if it must consist of at least 2

men and 2 women?

Solution

Step 1: Find the number of ways to choose 2 men and 3 women from the group.

There are (10

2)ways to choose 2 men from 10 men, and (8

3)ways to choose

3 women from 8 women.

Therefore, the number of ways to choose 2 men and 3 women is (10

2)×(8

3).

Step 2: Find the number of ways to choose 3 men and 2 women from the

group.

There are (10

3)ways to choose 3 men from 10 men, and (8

2)ways to choose

2 women from 8 women.

Therefore, the number of ways to choose 3 men and 2 women is (10

3)×(8

2).

Step 3: Add the results of Step 1 and Step 2 to find the total number of

ways to form the committee with at least 2 men and 2 women.

The total number of ways is (10

2)×(8

3)+(10

3)×(8

2).

Calculating these values, we get:

(10

2)= 45,(8

3)= 56,(10

3)= 120,(8

2)= 28

So the total number of ways to form the committee is 45 ×56 + 120 ×28 =

2520 + 3360 = 5880. Thus, the committee can be formed in 5880 ways.

Question 6

Question

A committee of 5 members is to be formed from a group of 8 men and 6 women.

In how many ways can the committee be formed if it must consist of at least 2

women?

Solution

Step 1: Calculate the number of ways to choose 2 women out of 6. There are

2)ways to choose 2 women from a group of 6. (6

2)=6!

2!(6−2)! = 15 ways.

Step 2: Calculate the number of ways to choose the remaining members from

the men and women. Since the committee must consist of at least 2 women, we

can choose the remaining members (men and/or women) from the remaining

total of 8 men and 4 women. We can choose the remaining 3 members from a

group of 8 men and 4 women in (12

3)ways. (12

3)=12!

3!(12−3)! = 220 ways.

Step 3: Find the total number of ways to form the committee. Multiply

the number of ways to choose the 2 women from Step 1 by the number of

ways to choose the remaining members from Step 2. Total number of ways =

15 ×220 = 3300.

Therefore, the committee can be formed in 3300 ways if it must consist of

at least 2 women.

Question 7

Question

In a particular city, there are 4 different train stations. A commuter needs

to travel from one station to another station for work. If the commuter al-

ways chooses the shortest route, how many different routes are possible for the

commuter to take?

Solution

Step 1: Since there are 4 different train stations, the commuter can choose the

starting station in 4 ways.

Step 2: After choosing the starting station, the commuter needs to choose

the destination station. Since the commuter always chooses the shortest route,

there is only 1 shortest route from any starting station to any destination station.

Step 3: Therefore, the total number of possible routes the commuter can

take is the product of the number of ways to choose the starting station and the

number of ways to choose the destination station.

Step 4: This gives us 4×1 = 4 possible routes that the commuter can take.

Hence, there are 4 different routes possible for the commuter to take when

traveling from one station to another station in the city.

Question 8

Question

In a group of 10 students, how many ways are there to select a committee of 5

students if exactly 2 of the students must be included?

Solution

Step 1: Calculate the number of ways to choose 2 out of the 10 students.

There are (10

2)ways to choose 2 students from a group of 10.

Step 2: Calculate the number of ways to choose the remaining 3 students

from the 8 students left after selecting the required 2 students.

There are (8

3)ways to choose 3 students from a group of 8.

Step 3: Multiply the results from Step 1 and Step 2 to find the total number

of ways to form the committee.

Multiplying (10

2)and (8

3)gives the total number of ways to choose a com-

mittee of 5 students.

Therefore, there are (10

2)×(8

3)ways to select a committee of 5 students where

exactly 2 students must be included.